Several Republican senators say President Barack Obama should keep up his charm offensive despite seeing his gun agenda shredded April 17 by a minority of the Senate.

That defeat added to the fizzle with which the president’s second term has begun, coming after Obama was unable to replace the budget sequester.

But Republican lawmakers said Obama’s effort to reach out to them could yet bear dividends as he tries to forge a bipartisan budget deal on entitlements and debt — and perhaps get an immigration overhaul he can sign.

“Guns are just an issue where senators have very strong views one way or the other on and charm’s not going to matter much,” said Sen. Lamar Alexander, R-Tenn.

Several Senate Democrats also rejected Obama’s appeals on the gun measures, fearing a general-election backlash from voters that Obama acknowledged April 17 in his Rose Garden comments. “Democrats had that fear, too. And so they caved to the pressure,” he said, admitting that the gun issue will have to be settled at the ballot box.

Alexander has been pushing the president for years to talk to lawmakers more and said he believes it can be effective.

“The president’s charm offensive is about the debt. I think he has been getting dividends on it. You can’t go four years with very little contact with members of the Senate and then change it overnight. But I think everyone I’ve talked to among Republican senators appreciates what he’s done the last two, three months,” he said.

“I feel like we’ve made some progress in our ability to do something about the debt between now and Aug. 1,” Alexander added. “I suspect that’ll pay dividends in other ways, such as the immigration bill.”

“I think on some of these issues like immigration and budget, the White House perspective might be a little more influential,” the Ohio Republican said. “They should continue to reach out and do more of it, and Republicans should be willing to sit and talk with them, especially given the fact the debt limit vote is coming up at the end of July. We’ve got to work something out.”

Sen. Jeff Flake, R-Ariz., who helped kill the gun measure despite personal pleas from former Rep. Gabrielle Giffords, D-Ariz., and her husband, Mark Kelly, is also one of the eight members of the bipartisan immigration overhaul group. He also said the defeat of the gun legislation shouldn’t be seen as a failure of the president’s charm offensive. “I still hope we can come to an agreement on the budget,” he said.

Sen. Patrick J. Toomey, R-Pa., a sponsor of the failed bipartisan compromise on expanded background checks for gun purchases, was less sanguine about reaching a deal on the budget, but he said that doesn’t mean the White House shouldn’t keep reaching out.

“One thing for sure is you never find common ground if you’re not having a conversation. ... I think we’re still very far apart. But I’m in favor of having conversations,” he said.

Still on the Move

Obama spokesman Josh Earnest also told reporters April 18 that the White House doesn’t see the gun defeat as affecting other pieces of his agenda, noting immigration as one area where legislation is moving forward.

“I don’t see any reason why anybody who’s taking a look at what’s happening in Congress right now should despair about the possibility for bipartisanship,” he said. “I think what the president is frustrated by, in reaction to the gun vote yesterday, was a willingness to willfully distort the facts that were included in that legislation, and as I mentioned, the outsized influence that’s being exercised by some special interests in Washington, D.C., and the unwillingness of some members of the United States Senate to stand up to them.”

As for Obama’s rebuke April 17 to the senators who blocked the gun bill — calling it a “shameful” day in Washington and one where lawmakers were intimidated by a lying gun lobby, Alexander brushed it off. “I think he’s entitled to express himself,” he said.

There remain big hurdles to any budget deal that meets the president’s requirement that it include new tax revenue from the wealthy — something GOP leaders insist they will not accept.

That could mean Obama’s hopes for a bold second-term agenda may rest with immigration, the one issue where a big, bipartisan compromise has been proposed with the support of leaders in both parties.

That momentum springs from the electoral drubbing Republicans suffered among Hispanics in last year’s elections and their interest in avoiding a repeat in 2014.